The concept of impedance extends also to masses and springs.
Figure 7.2 illustrates an ideal mass of
kilograms
sliding on a frictionless surface. From Newton's second law of motion, we
know force equals mass times acceleration, or

This is the transfer function of an integrator. Thus, an ideal mass
integrates the applied force (divided by
) to produce the output
velocity. This is just a ``linear systems'' way of saying force equals
mass times acceleration.

Since we normally think of an applied force as an input and the resulting
velocity as an output, the corresponding transfer function is
. The system diagram for this view
is shown in Fig. 7.3.

The impulse response of a mass, for a force input and velocity output,
is defined as the inverse Laplace transform of the transfer function:

In this instance, setting the input to
corresponds to
transferring a unit momentum to the mass at time 0
. (Recall that
momentum is the integral of force with respect to time.) Since momentum is
also equal to mass
times its velocity
, it is clear that the
unit-momentum velocity must be
.

Figure 7.3:
Input/output description of a
general impedance, with force
as the input, velocity
as
the output, and admittance
as the transfer function.

Once the input and output signal are defined, a transfer function is
defined, and therefore a frequency response is defined [487].
The frequency response is given by the transfer function evaluated on
the
axis in the
plane, i.e., for
. For the ideal mass,
the force-to-velocity frequency response is

Again, this is just the frequency response of an integrator, and we can
say that the amplitude response rolls off dB per octave, and the phase
shift is
radians at all frequencies.

In circuit theory, the element analogous to the mass is the inductor,
characterized by
, or
. In an analog
equivalent circuit, a mass can be represented using an inductor with value
.